2023/09/18 by Alberto Ohashi, Ohashi, Alberto, Francesco Russo +3 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2309.09871
openalex publication_date 2023/09/18 · openalex created_date 2023/09/20 · openalex updated_date 2026/08/01
In this work, we present a detailed analysis on the exact expression of the L2-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion B with parameter (1)/(4) < H < (1)/(2). Our main result is a complete description of a Hilbert space of integrand processes which realizes the L2-isometry where none regularity condition in the sense of Malliavin calculus is imposed. The main idea is to exploit the regularity of the conditional expectation of the tensor product of the increments Bt-δ,t+δ⊗ Bs-ε,s+ε onto the Gaussian space generated by (Bs,Bt) as (δ,ε)\downarrow 0. The Hilbert space is characterized in terms of a random Radon σ-finite measure on [0,T]2 off diagonal which can be characterized as a product of a non-Markovian version of the stochastic Nelson derivatives. As a by-product, we present the exact explicit expression of the L2-norm of the pathwise rough integral in the sense of Gubinelli.